How to make non-linear movements with a Follow Path constraint?
The Evaluation Time property is keyframable, you can shape its F-curve to any function you like.
Select your bezier curve object, in the graph editor and expand it to show the Evaluation Time curve (add a keyframe if necessary if it doesn't show by default).
Under the Modifier panel add a Generator modifier, change poly order to $2$, and adjust the function to get the desired speed.